Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 097, 22 pages      arXiv:0910.3609      https://doi.org/10.3842/SIGMA.2009.097
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

Christine Scharlach
Technische Universität Berlin, Fak. II, Inst. f. Mathematik, MA 8-3, 10623 Berlin, Germany

Received May 08, 2009, in final form October 06, 2009; Published online October 19, 2009

Abstract
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(TpM) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S = HId (and thus S is trivially preserved). In Part 1 we found the possible symmetry groups G and gave for each G a canonical form of K. We started a classification by showing that hyperspheres admitting a pointwise Z2 × Z2 resp. R-symmetry are well-known, they have constant sectional curvature and Pick invariant J < 0 resp. J = 0. Here, we continue with affine hyperspheres admitting a pointwise Z3- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z3) or quadrics (SO(2)) with a curve.

Key words: affine hyperspheres; indefinite affine metric; pointwise symmetry; affine differential geometry; affine spheres; warped products.

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